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\title{\vglue -20pt \huge \bf The Viviani Curve \footnote{This file is from the 3D-XploreMath project. \hfil\break Please see http://vmm.math.uci.edu/3D-XplorMath/index.html}}
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\LARGE

The Viviani curve is the intersection of a sphere  of radius $2\ aa$
and a cylinder of radius $aa$ that touch at a single point, 
the double point of the curve. Parametric formulas for it are:

$z = aa\ (1+\cos(t))   = aa\ 2\ \cos(t/2)^2,$

$y = aa\ \sin(t)           = aa\ 2\ \sin(t/2)\ \cos(t/2),$ and

$x = aa\ 2\ \sin(t/2)$

Implicit equations for the two intersecting surfaces are:

 $x^2 + y^2 + z^2 = 4\ aa^2$,    a sphere of radius $2\ aa$, and 

 $(z-aa-bb)^2  + y^2  = aa^2$,     a cylinder of radius $aa$.

The planar projections of this curve are therefore in
general curves of degree 4, but because of its symmetries 
the Viviani curve has two orthogonal two-to-one projections
that are simpler; namely curves of degree 2. Indeed 
projecting it to the y-z-plane we get a twice covered circle 
(use Settings Menu: Set Viewpoint and Up Direction  200,0,0),
projecting to the x-z-plane gives a twice covered
parabolic piece, $(1 - z/(2aa)) = (x/(2aa))^2$, while
the projection to the x-y-plane is the degree 4 figure 8
with the equation (for $aa=1/2$): $x^2 - y^2 = x^4$.

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